= Solution
Use the upper-index coordinates $B^\mu,B^{*\mu}$ specified in the paper. Differentiation of the <Lagrangian density> gives the <canonical momenta>
$$
\boxed{\pi_\mu=-G^{*0}{}_{\mu},\qquad\pi_\mu^*=-G^0{}_{\mu},\qquad\pi_0=\pi_0^*=0.}
$$
In particular, $\pi_i=G^{*0i}$ and $\pi_i^*=G^{0i}$. No time derivative of $B^0$ appears, so the zero momenta are primary <constraints>, not invertible equations for $\dot B^0$.
For spatial indices, $G^{0i}=\dot B^i+\partial_iB_0$, so $\dot B^i=\pi_i^*+\partial^iB_0$, with the conjugate relation for $B^{*i}$. Also
$$
\mathcal L=\sum_i|G^{0i}|^2-\frac12G_{ik}^*G^{ik}+M^2|B_0|^2-M^2\sum_i|B^i|^2.
$$
The <Legendre transform> for the <complex Proca Hamiltonian>, $\mathcal H_c=\pi_i\dot B^i+\pi_i^*\dot B^{*i}-\mathcal L$ therefore gives
$$
\boxed{H_c=\int d^3x\left[\pi_i^*\pi_i+\pi_i\partial^iB_0+\pi_i^*\partial^iB_0^*+\frac12G_{ik}^*G^{ik}-M^2B_0^*B_0-M^2B_i^*B^i\right].}
$$
Repeated spatial indices in the positive product $\pi_i^*\pi_i$ denote an ordinary three-component sum. A total <Hamiltonian> additionally includes multipliers for $\pi_0=\pi_0^*=0$.
The Hamiltonian printed in the source cannot follow from the given complex <Lagrangian density>: its missing conjugations, mass coefficients and spatial mass sign are inconsistent. A concrete check is spatially uniform initial data $B^1=b\in\mathbb R$, $B^0=B^2=B^3=0$, with all <canonical momenta> zero. The correct <Hamiltonian density> is $M^2b^2$, while the printed expression gives $-M^2b^2/2$. These data satisfy the initial <constraints> and evolve as a spatially uniform massive oscillator. Thus the discrepancy is not a harmless surface term or a different complex-field normalization.
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